The figure below shows an “unfolded” view of a typical 6-sided die (d6), used everywhere from board games to casinos:
Each of the six faces is uniquely labeled with the integers 1 through 6. But that’s not all; note that the values on opposite faces always sum to 7. This is a standard arrangement applied more generally to other types of dice as well. My set of Platonic solid dice– with sides for — all have this same property. Let us call such an -sided die standard if all opposite faces sum to .
Problem 1: How many different standard 6-sided dice are there? That is, in how many ways can we label the faces with distinct integers 1 through 6, with opposite faces summing to 7?
Problem 2: What if we relax the standard (constant opposite sum) property? That is, in how many ways can we label the faces with distinct integers 1 through 6, with no other restrictions?
Problem 3: Same as Problem 2, but for the other Platonic solids (d4, d8, d12, d20)?
I think these are great examples of problems for students that straddle whatever line there may be between mathematics and computer science. Problems 1 and 2 can be solved as-is “by hand.” (And they are interesting in part because the answers are perhaps surprisingly small numbers.) But the usual mathematical machinery involved just counts the number of dice in each case; it’s an interesting extension as a programming problem to actually enumerate (i.e., list) them, display visual representations of them, etc.
Problem 3 is more challenging; the machinery is the same, but the larger numbers involved require some amount of automation in the housekeeping.
Now for what I think is a really hard problem motivating this post (Edit: although after a response from a reader in another forum with a very elegant solution, perhaps this is not as difficult as I thought it might be!):
Problem 4: Same as Problem 1, but for the other Platonic solids– that is, how many different standard (d4, d8, d12, d20) are there, with constant opposite sums?